Schikhof's Ultrametric Calculus, originally published in 1984, is now back as a (print-on-demand) paperback. It is still a very useful book.

Books on p-adic numbers and p-adic analysis usually reflect quite closely the interests and concerns of their authors. Number theorists tend to emphasize the algebraic side and to focus on results that show up in their work: the local-global principle, functions defined by power series, p-adic interpolation, p-adic measures. Those interested in the p-adic version of the theory of analytic manifolds (usually called "rigid geometry") have a completely different set of concerns, which dominate, for example, Non-Archimedean Analysis, by Bosch, Guntzer, and Remmert.

Schikhof's book is written mostly from the point of view of analysis. After doing the necessary preliminary work of building the p-adic numbers and other fields with non-archimedean absolute values, the book repeatedly goes through the same steps: here's a theorem or set of theorems from analysis; does it work for non-archimedean fields?; if so, can we strengthen it?; if not, can we modify it so that it does?

Take, for example, basic differential calculus. Derivatives can be defined in the same way as usual, but much of the standard theory does not work. The reason, ultimately, is that the Mean Value Theorem is not true, resulting in such monsters as functions whose derivative is everywhere zero but which are not constant (not even locally constant). Most books (including mine!) get to that point, decide that this means there is nothing to study, and give up. Schikhof instead formulates a definition of C1 functions that actually yields a usable theory. He ends up developing a sophisticated theory of Cn functions. This includes exploring the relations between the class of Cn functions and the better-known class of analytic functions.

Schikhof's discussion of integration is particularly nice. He discusses the options, shows that some are not useful (there is no non-trivial bounded translation-invariant additive Qp-valued measure on Qp, for example) and shows that others do work, though we end up with a wide range of incompatible variants in the end. He also looks at anti-derivatives, which are unrelated to integrals and have a much more interesting theory than in the classical case (in part because there are so many "pseudoconstant" functions, whose derivatives are zero).

Some of the material seems to be Schikhof's own work. For example, he introduces a notion of "increasing" functions, and generalizes it to "monotone" functions of several types. He ends up with a notion of "sign" (or "side of zero") that yields more than two "signs", and hence more than two kinds of "monotone" functions. It's not clear whether these notions will ever be useful (as Schikhof himself admits), but it is certainly interesting that such an idea can be made to work.

Number theorists will find much of interest here as well. Schikhof gives constructions of p-adic Gamma and Zeta functions and develops enough integration theory to allow the reader to follow the measure-theoretic approach to p-adic L-functions. He even does a little bit with p-adic differential equations.

Though the book is labelled "an introduction," the pre-requisites are quite heavy. Schikhof assumes that the reader knows quite a bit of (classical) analysis, general topology, and the basics of normed vector spaces and Banach spaces. He also assumes knowledge of abstract algebra (for example, completions are constructed, on page 15, by taking the quotient of the ring of all Cauchy sequences by the maximal ideal of null sequences). Thus, the book is best thought of, at least in the American context, as being aimed at graduate students who already have good control of fundamental mathematics.

A lot of interesting stuff is in the exercises, some of which seem quite hard. A solutions manual would be very welcome, particularly for those of us who might want to refer to these results! (I consider it a faux pas to cite an exercise when one needs a result… but there are a lot of nice results that appear as exercises here.)

The photographic reproduction is adequate; all the funkiness is in the original typesetting. (Even the cover is funky: who exactly decided that rotating the "p" a little bit suggests the usual italic p?) The English is occasionally strange. One of my favorites is the way Schikhof uses "apparently" to mean "anyone can see that." It brought me up short every time, and made me smile.

All in all, this is a very useful book, a nice complement for books such as Katok's p-adic Analysis Compared with Real. Those who enjoy that book and want to know more can find quite a bit more here. For number-theoretically minded users of p-adic analysis, Schikhof's book will serve as a very useful reference, a place to go for careful definitions, useful theorems, and pointers to important results.

Is Liouville's theorem true for p-adic analytic functions? Schikhof's book is the place to find out. It is good to have it available once again.

Fernando Q. Gouvêa is the editor of MAA Reviews and the author of p-adic Numbers: An Introduction.